Repetition rate

The repetition rate $f_\text{rep}$ of a pulsed laser is the number of pulses emitted per second, and its inverse is the pulse period. For a mode-locked oscillator it is fixed by the cavity: one pulse circulates, and each time it strikes the output coupler a fraction leaves, so pulses emerge once per round trip. A Ti:sapphire oscillator with a 1.87 m linear cavity runs at 80.2 MHz, a period of 12.5 ns. Q-switched lasers, regenerative amplifiers and pulsed fiber amplifiers have a repetition rate set by electronics instead, anywhere from single shots to several megahertz.

Cavity round trip

For a linear (standing-wave) cavity of optical length $L$, the round-trip time is $2L/c$ and

$$f_\text{rep} = \frac{c}{2L}.$$

A 1.5 m cavity in air gives 99.9 MHz, a 10.0 ns period; conversely, 80 MHz requires $L$ = 1.874 m, which is folded into a compact box with several curved mirrors. In a ring cavity the pulse passes once per round trip, so $f_\text{rep} = c/(n_g L)$ with $L$ the ring circumference and $n_g$ the group index. A fiber ring with 10 m of silica fiber ($n_g \approx$ 1.47) runs at 20.4 MHz, and reaching about 100 MHz requires a total length of only 2 m. The repetition rate equals the longitudinal mode spacing, which is why the pulse train of a mode-locked laser is also a comb of optical frequencies spaced by $f_\text{rep}$.

Typical ranges

Semiconductor and microresonator sources reach tens of gigahertz because their cavities are millimetres or less. At the other end, high-energy Nd:glass and Ti:sapphire systems for plasma physics fire at 10 Hz or once per several minutes, limited by heat removal from the gain medium.

Duty cycle and energy

The repetition rate links the three quantities most often quoted for a pulsed laser. The pulse energy is $E = P_\text{avg}/f_\text{rep}$, and the duty cycle is $\tau f_\text{rep}$ for pulse duration $\tau$. A 100 fs oscillator at 80 MHz has a duty cycle of 8 × 10⁻⁶; a 10 ns Q-switched laser at 100 kHz has 10⁻³. The ratio of peak power to average power is approximately the reciprocal of the duty cycle. The pulsed laser calculator evaluates all three from any two measured inputs plus the pulse duration.

In a Q-switched laser the pulse energy falls when the repetition rate exceeds roughly the inverse of the upper-state lifetime, because the gain medium has less time to store energy between pulses. For Nd:YAG, with a 230 µs lifetime, that crossover is near 4.3 kHz: above it the average power levels off and the pulse energy drops roughly as $1/f_\text{rep}$, and the pulses lengthen.

Changing the repetition rate

The oscillator rate can be tuned only slightly, by moving a cavity mirror on a translation stage or piezo. Larger reductions use a pulse picker: a Pockels cell between polarizers, or an acousto-optic modulator, opens for one pulse in every $N$. Picking 1 MHz from an 80 MHz train keeps one pulse in 80, so the pulse energy stays 12.5 nJ for a 1 W oscillator while the average power falls to 12.5 mW. Amplifiers then raise the energy of the selected pulses, as in a regenerative amplifier. Cavity dumping is the alternative: it extracts most of the circulating pulse energy at a reduced rate instead of discarding pulses.

Measurement

A fast photodiode feeding a frequency counter or RF spectrum analyzer gives $f_\text{rep}$ directly; the spectrum analyzer shows the fundamental and its harmonics, and the width and sidebands of these lines measure timing jitter. An oscilloscope shows the pulse period and reveals missing or doubled pulses.

Pitfalls

Harmonic mode locking, common in fiber lasers, places several equally spaced pulses in the cavity and multiplies the repetition rate by an integer; an oscilloscope trace at the expected period is the check. Q-switched mode locking produces bursts of mode-locked pulses under a slower envelope, and a counter may lock to either rate. Pulse pickers with finite contrast pass weak neighbouring pulses, which amplifiers can raise to a significant fraction of the main pulse.

Common questions

How does cavity length set the repetition rate?

The pulse leaves once per round trip, so $f_\text{rep} = c/2L$ for a linear cavity; doubling the length halves the rate.

Does a higher repetition rate give more power?

At fixed pulse energy the average power scales with $f_\text{rep}$. In practice the average power is limited by pump power and heat, so raising the repetition rate usually lowers the energy per pulse.

References: A. E. Siegman, Lasers (University Science Books, 1986); U. Keller, Recent developments in compact ultrafast lasers, Nature 424, 831 (2003); W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006); B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).